SAT Mc Graw Hill 2011

The Laws of Arithmetic
When evaluating expressions, you don’t always have to
follow the order of operations strictly. Sometimes you
can play around with the expression first. You can
commute(with addition or multiplication), associate
(with addition or multiplication), or distribute(multi-
plication or division over addition or subtraction).
Know your options!

When simplifying an expression, consider

whether the laws of arithmetic help to make it

easier.

Example:

57(71) +57(29) is much easier to simplify if, rather

than using the order of operations, you use the

“distributive law” and think of it as 57(71 +29) =

57(100) =5,700.

The Commutative and
Associative Laws

Whenever you add or multiply terms, the order

of the terms doesn’t matter, so pick a convenient

arrangement. To commute means to move

around. (Just think about what commuters do!)

Example:

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 =

1 + 9 + 2 + 8 + 3 + 7 + 4 + 6 + 5

(Think about why the second arrangement is

more convenient than the first!)

Whenever you add or multiply, the grouping of

the terms doesn’t matter, so pick a convenient

grouping. To associate means to group together.

(Just think about what an association is!)

Example:

(32 ×4) ×(25 ×10) ×(10 ×2) =

32 ×(4 ×25) ×(10 ×10) × 2

(Why is the second grouping more convenient

than the first?)

Whenever you subtract or divide, the grouping

of the terms does matter. Subtraction and divi-

sion are neither commutative nor associative.

Example:

15 − 7 − 2 ≠ 7 − 15 − 2 (So you can’t “commute”

the numbers in a difference until you convert it to

addition: 15 +− 7 +− 2 =− 7 + 15 +−2.)

24 ÷ 3 ÷ 2 ≠ 3 ÷ 2 ÷ 24 (So you can’t “commute”

the numbers in a quotient until you convert it to

multiplication:

The Distributive Law

When a grouped sum or difference is multiplied

or divided by something, you can do the multipli-

cation or division first (instead of doing what’s in-

side parentheses, as the order of operations says)

as long as you “distribute.” Test these equations

by plugging in numbers to see how they work:

Example:

a(b+c) =ab+ac

Distribution is never something that you have

to do. Think of it as a tool, rather than a re-

quirement. Use it when it simplifies your task.

For instance, 13(832 + 168) is actually much

easier to do if you don’t distribute: 13(832 +

168) = 13(1,000) = 13,000. Notice how annoy-

ing it would be if you distributed.

Use the distributive law “backwards” when-

ever you factor polynomials, add fractions, or

combine “like” terms.

Example:

9 x^2 − 12 x= 3 x(3x−4)

Follow the rules when you distribute! Avoid

these common mistakes:

Example:

(3 +4)^2 is not 3^2 + 42

(Tempting, isn’t it? Check it and see!)

3(4 ×5) is not 3(4) ×3(5)

57 27 37−=

33

b

a

b

a

b

+=

+

bc

a

b

a

c

a

()+

=+

24

1

3

1

2

1

3

1

2

××=×× 24 .)

Lesson 2: Laws of Arithmetic

274 MCGRAW-HILL’S SAT